Our posets play a pivotal role in the study of bichains and graphs. If a poset \(P:= (V, \le )\)  is coverable by two chains, then the order \(\le \)  is the intersection of two linear orders \(\le _1\)  and \(\le _2\)  on \(V\) . Let us call bichain every structure \(B:= (V, (\le _1, \le _2))\)  made of a set \(V\)  and two linear orders \(\le _1\)  and \(\le _2\) ; denote by \(o( B)\)  the poset \((V, \le _1\cap \le _2)\) . Then each of our posets is of the form \(o(B)\)  for some bichain. Finite bichains are a way of looking at (finite) permutations. In fact, each bichain \(B\)  on a \(n\) -element set can be coded in a unique way by a permutation \(\sigma \)  of \(\lbrace 1, \dots , n\rbrace \)  (as a \(n\) -element chain can be coded by the natural order on \(\lbrace 1, \dots n\rbrace \) ). As observed by Cameron [1]}, the embeddability between bichains corresponds to an order on the set \(\Omega \)  of permutations of \(\lbrace 0, \dots , n-1\rbrace \) , for \(n\in {\mathbb {N}}\) , an order previously defined in relation with the Stanley-Wilf Conjecture. Since this conjecture was solved (by Marcus and Tardos [2]} 2004), hereditary classes of permutations (defined as classes \(\mathfrak {C}\)  de permutations \(\sigma \)  such that \(\sigma \le \tau \in \mathfrak {C} \)  implies \(\sigma \in \mathfrak {C}\) ) and their generating series have been intensively studied (see [3]} for an overview). Due to Cameron's observation, their study fits in the study of hereditary classes of finite relational structures.
Graphs come in the picture too. To every poset \(P\) , one may associate the incomparability graph of \(P\) , denoted \(Inc(P)\) , and to a class \(\mathfrak {C}\)  of posets the class \(Inc<\mathfrak {C}>:= \lbrace Inc(P): P \in \mathfrak {C}\rbrace \) . The incomparability graphs of these finite posets are in fact bipartite permutation graphs. With bichains and graphs associated to our posets comes a natural question: how does the embeddability between posets relate to the embeddability between the associated bichains and graphs? We have only few elements of answer. We may note that if \(\mathcal {B}\)  is a w.q.o. class of bichains then, the class \(o<\mathcal {B}>:=\lbrace  o(B): B\in \mathcal {B}\rbrace \)  is w.q.o. Also, if \(\mathfrak {C}\)  is w.q.o. then \(Inc<\mathfrak {C}>\)  is w.q.o. too. It is not known if the reverse of these implications holds. That is, if \(\mathfrak {C}\)  is a w.q.o. class of posets, for which the order is intersection of two linear orders, is the class \(o^{-1}<\mathfrak {C}>: = \lbrace  B: o(B)\in \mathfrak {C}\rbrace \)  is w.q.o? And if \(Inc<\mathfrak {C}>\)  is w.q.o, is \(\mathfrak {C}\)  w.q.o.? We prove that the reverse of these implications hold if \(\mathfrak {C}\)  is a hereditary class of posets coverable by two chains. As stated in Theorem REF  below, parts of these implications hold in a more general situation.
